Trapezium-Type Inequalities for an Extension of Riemann–Liouville Fractional Integrals Using Raina’s Special Function and Generalized Coordinate Convex Functions

: In this paper, the authors analyse and study some recent publications about integral inequalities related to generalized convex functions of several variables and the use of extended fractional integrals. In particular, they establish a new Hermite–Hadamard inequality for generalized coordinate φ -convex functions via an extension of the Riemann–Liouville fractional integral. Furthermore, an interesting identity for functions with two variables is obtained, and with the use of it, some new extensions of trapezium-type inequalities using Raina’s special function via generalized coordinate φ -convex functions are developed. Various special cases have been studied. At the end, a brief conclusion is given as well.

This property is defined in the following works of Jensen J.L.W.V. (1905 and 1906) [26,27] as follows. Definition 1. A function f : I ⊆ R −→ R is said to be convex on I, if: holds for every 1 , 2 ∈ I, and ı ∈ [0, 1]. This property is a necessary condition for the classical Hermite-Hadamard inequality, which is established as follows.
Theorem 1. Let f : I ⊆ R −→ R be a convex function on I and 1 , 2 ∈ I with 1 < 2 . Then, the following inequality holds: This inequality (6) is also known as the trapezium inequality.
Furthermore, several papers have also been published that relate integral inequalities to fractional calculus and special functions [14,28]. In [29], Sambandham, S. wrote: "the advantage of using fractional derivative versus the integer derivative is that the integer derivative is local in nature, where as the fractional derivative is global in nature"; this notion invites us to think about the behaviour of generalized convex functions in the setting of integral inequalities of fractional order.
Given the introduction of an extension of the Riemann-Liouville fractional integral made by Awan [30] and the relevance of the Hermite-Hadamard inequality in the field of statistics and probability theory, which in turn involves all the research in applied science, the purpose of the present work is to establish some integral inequalities of the trapezium type using this type of double integral for generalized convex functions in coordinates.

Preliminaries
Following the notation used by Dragomir S.S. in [8], we recall the following definition. Let us consider the rectangle ∆ = [ 1 , 2 ] × [ 3 , 4 ] ⊂ R 2 with 1 < 2 and 3 < 4 . A function f : ∆ −→ R is said to be convex on ∆ if the following inequality holds: for all (x, y), (z, w) ∈ ∆ and ı ∈ [0, 1]. A function f : ∆ −→ R is said to be convex on coordinates ∆ if the partial functions f y : In the work of Awan et al. [30], the following definition used in the study of a two-dimensional extension of the Hermite-Hadamard inequality was found.
Noor M.A. in [16] introduced the concept of φ-convex function with the assumption that K is a non-empty closed set in R n and φ : K → R a continuous function. Definition 3. Let u ∈ K. If there exists a function φ such that the set K is said to be a φ-convex set: for all 1 , 2 inK and t ∈ [0, 1], then the set K is usually called a φ-convex set.

Definition 4.
Given a function f : then the function is called φ-convex.
Note that every convex function is φ-convex, but the converse does not hold in general.
Awan M.U. et al. in [30] defined some new extensions for fractional integrals.
From the above definition and fixing the mean value between the extremes of the intervals, we have: Motivated by the aforementioned literature, the paper is organized as follows: In Section 3, a new Hermite-Hadamard inequality for generalized functions in Definition 7 via the Riemann-Liouville fractional integral will be established. Furthermore, an interesting identity for functions with two variables will be given. By using the established identity, some new extensions of trapezium-type inequalities for Raina's fractional integral operators via generalized coordinate φ-convex functions and some special cases will be obtained. In Section 4, a brief conclusion will be provided as well.

Main Results
Our first result is the Hermite-Hadamard inequality for generalized coordinate φ-convex functions via the Riemann-Liouville fractional integral.
To derive our second results, we establish a new integral identity for the partial differentiable function involving Raina's functions.
If ∂ 2 f ∂t∂r ∈ L 1 ( ), then the following identity holds: where: Proof. Let: Similarly, we have: and: By using the change of variables, we have: Multiplying Equality (16) by , we get the desired equality (15). The proof is complete.
Using Lemma 1, we can derive the following theorems for generalized coordinate φ-convex functions.
∂t∂r q is a generalized coordinated φ-convex function where q > 1 and 1 p + 1 q = 1, then the following inequality holds: Proof. Using Lemma 1, the fact that is a generalized coordinated φ-convex function, and Hölder's inequality, we have: The proof is complete.
We point out some special cases of Theorem 3.
Proof. Using Lemma 1, the fact that ∂ 2 f ∂t∂r q is a generalized coordinated φ-convex function, and the well-known power-mean inequality, we have: